L2-3_Algebra.fm Page 59 Tuesday, August 19, 2003 2:49 PM
Algebra
Patterns in Division
Math Concepts
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Overview
division
fractions
whole numbers
decimals
Students will investigate how the quotients and remainders
from integer division relate to the quotients from rational
number division in both fraction and decimal form.
Materials
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TI-15 Explorer™
Patterns in Division
recording sheets
objects for counters
pencils
Introduction
1. Pose a simple problem like this one: If you had 10 cookies to divide
evenly among 3 people, how many cookies would each person get?
Have students act out the division with counters. (Or, use the cookie
pictures provided on page 63.)
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Ask: What would you do with the leftover cookie? Discuss the two
choices: Leave the cookie alone and let it be left over, or cut it into
three equal pieces to give to the three people.
2. Discuss how you might record the first choice with a whole number
quotient and a whole number remainder and how you might record
the second choice with a fraction in the quotient.
3. Demonstrate using the integer divide key £ on the calculator to
record the situation in which you leave the extra cookie alone:
10 £ 3 Z to display 3 r 1.
4. Have students compare this method of division to dividing in
fraction notation, displaying the quotient in mixed number form.
Enter 10  3 to represent 10 divided by 3 as a fraction. Then press
® to display 3 1--- , the mixed number quotient for 10 divided by 3.
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5. Finally, have students change the fraction to decimal form using
Ÿ.
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How would changing the
Fraction Display Mode affect
your result? ¢ "®
M
3
Use the scroll feature, > ?,
to compare results.
6. Have students compare this result to the decimal quotient found
by entering 10 W 3 Z.
7. Have students record their data on their recording sheets.
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L2-3_Algebra.fm Page 60 Tuesday, August 19, 2003 2:49 PM
Algebra
Patterns in Division (continued)
Introduction (continued)
8. Ask students: What data do you think you would have collected if
you had started with a different number of cookies? What if there
had been 20 cookies instead of 10? 30 cookies? 35 cookies?
9. Have students work in small groups to investigate what happens
when they use 4 as a divisor. Have them record the data on their
recording sheets, try other divisors, and look for patterns in the data.
Collecting and Organizing Data
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While students generate data for the different divisors, ask questions
such as:
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What divisor are you using? What kinds of remainders do you
expect to find? (Refer to the Recurring Remainders activity
on page 11.)
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What does the quotient represent? What does the remainder
represent?
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What does the fraction form of the quotient represent?
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What patterns do you notice in the fraction form of the quotients?
How do the fractions relate to the remainders? To the divisors?
Why do you think this happens?
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What happens when you change the divisor? How do
your remainders and fractions change? How do their
relationships change?
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What relationships do you see between the fractions and
the decimals?
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What kind of conjectures can you make about relationships
between the remainders and the fractions? The relationships
between the fractions and the decimals?
60
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How are the different ways
to perform division on the
calculator alike? How are they
different? Do they differ from
calculator to calculator?
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L2-3_Algebra.fm Page 61 Tuesday, August 19, 2003 2:49 PM
Algebra
Patterns in Division (continued)
Analyzing Data and Drawing Conclusions
After students have made and compared several tables and looked for
relationships, have them discuss their results as a whole group.
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When would you most likely
want to use £?
Ask questions such as:
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When would you want to use W
on the calculator?
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When might you want to
represent division as a fraction
with  on the calculator?
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What kinds of remainders occur with a divisor of ______? Why?
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What kinds of fractions occur in the quotients with a divisor of
______? Why?
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What kinds of decimals occur in the quotients with a divisor of
______? Why do you think those decimals look the way they do?
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Do the related fraction quotients and decimal quotients represent
the same amount? Why do you think this is true or not true?
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Now what kinds of questions do you have about division,
remainders, fractions, and decimals? How might we investigate
these questions?
© 2003 Texas Instruments Incorporated. ™Trademark of Texas Instruments Incorporated.
Continuing the Investigation
Have students collect data about other divisors to see whether their
conjectures continue to be true or whether the patterns in their data lead
them to new conjectures.
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Algebra
Name:
Patterns in Division
Recording Sheet
Collecting and Organizing Data
Texas Instruments grants teachers permission to reproduce this page for use with students. © 2003 Texas Instruments Incorporated.
# of Cookies
(Dividend)
# of People
(Divisor)
Integer Quotient
and Remainder
Fraction
Quotient
Decimal
Quotient
Analyzing Data and Drawing Conclusions
After you have collected the data in the chart above, discuss this exercise as a group and record your
findings. Enter any questions you come up with in the space below, and be prepared to discuss them.
Questions we thought of while we were doing this activity:
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Uncovering Mathematics with Manipulatives, the TI-10, and the TI-15 Explorer™ Calculator
L2-3_Algebra.fm Page 63 Tuesday, August 19, 2003 2:49 PM
Algebra
Patterns in Division
Texas Instruments grants teachers permission to reproduce this page for use with students. © 2003 Texas Instruments Incorporated.
Cookies
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